Most physical systems are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated
simultaneously with the system's equations. We propose splitting procedures featured by a SB3A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.
We consider splitting methods for the numerical integration of non-autonomous sep-
arable differential equations. Splitting methods have been extensively used as geometric numerical
integrators during the last years showing excellent performances (both qualitatively and quantita-
tively) when applied on many problems. They are designed for autonomous separable systems and
a substantial number of methods tailored for different structures of the equations have recently ap-
peared. When these methods are used on non-autonomous problems, usually their performance
diminishes considerably, and even they can lose the order of accuracy. Previous attempts to preserve
such performance on non-autonomous problems required to modify, in a non-trivial way, the existing
methods. In this paper, we present a simple alternative which, in many relevant cases, allows to
retain the high performance of the splitting methods using the same schemes as for the autonomous
problems. This technique is applied on different problems and its performance is illustrated on several
numerical examples.
A partire da fenomeni reali viene esposto il percorso che, mediante il linguaggio della matematica, conduce all'astrazione ed alle teorie assiomatiche formali.
modellistica matematica
assiomatizzazione della realtà
linguaggio matematico
Capobianco, Criscuolo and Junghanns [2] have studied an integro differential equation of
Prandtl type
and a collocation method as well as a quadrature method for its approximate solution in weighted Sobolev spaces. Furthermore,
collocation and collocation quadrature methods for the same integral
equation have been studied in weighted spaces of continuous functions
[3]. The aim of the present paper is to present an algorithm related
to a numerical model for a hypersingular Integral equation arising in a solid circular plate problem, based on the collocation
methods with quadrature methods on orthogonal polynomials as in [2, 3]. The optimal convergence rates is proved.
The analysis of the convergence and stability theory is carried out for a class of integrals
accessible to Gauss quadrature as singular principal value integrals. Numerical examples confirming the theoretical results are
also given.
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